Look at the diagram you populated last time with all sets from P$($A$),$ with A $=\{1,2,3\}.$

$($It$$s called a Hasse diagram, named after the algebraist Helmut Hasse.$)$ It is a pictorial

representation of two important mathematical concepts: first, partial order, and, second,

a Boolean algebra. A Boolean algebra is a structure $($or domain$-$plus$)$ D$.$ It consists of a

set, say A$,$ with two distinguished elements $>($top$)$ and $($bottom$).$ Moreover, the set

A is equipped with two binary operations: $($meet$)$ and $($join$),$ and a unary operation

$($complement$).$ The binary operations are well$-$behaved in familiar ways: they are each

commutative and associative as well as distributive when they both interact. But they also

have other properties, namely, for all elements a$,$ b in A$,$ we have:

$($i$)$ a $($a $$ b$)=$ a a $($a $$ b$)=$ a absorption

$($ii$)$ a $>=$ a a $=$ a identity

$($iii$)$ a $$ a $=$ a $$ a $=>$ complement

Show that the the powerset P$($A$)$ ordered by the relation $($as in your diagram from last

week$$s assignment$)$ forms a Boolean algebra $($a$)$ by identifying the three Boolean operations

$($meet$),($join$),$ and with set$-$theoretic operations you know, and $($b$)$ by identifying $>$

$($top$)$ and $($bottom$)$ with specific subsets of A in such a way that the three laws $($i$)($iii$)$

above come out true. $[2$PBoolean algebra $($of sets$).$ Let $A$ be a set. Then the following assignments will work when we consider all

subsets of $A$ :

interpret the Boolean operation ${?}^{{?}^{?}}($meet$)$ as the set$-$theoretic operation $($union$)$

interpret the Boolean operation ${?}^{{?}^{?}}($meet$)$ as the set$-$theoretic operation $n($intersection$)$

interpret the Boolean operation ${?}^{{?}^{?}}($meet$)$ as the set$-$theoretic operation $\frac{\frac{?}{?}}{?}($difference$)$

interpret the Boolean operation $vvv($join$)$ as the set$-$theoretic operation $($union$)$

interpret the Boolean operation $vvv($join$)$ as the set$-$theoretic operation $($intersection$)$

interpret the Boolean operation $vvv($join$)$ as the set$-$theoretic operation $\frac{\frac{?}{?}}{?}($difference$)$

interpret the Boolean operation $-($complement$)$ as the set$-$theoretic operation $-($complement$)$

interpret the Boolean operation $-($complement$)$ as the set$-$theoretic operation $P($powerset$)$

interpret the Boolean constant ${?}^{TT}($top$)$ as the empty set.

interpret the Boolean constant ${?}^{TT}($top$)$ as the set A itself.

interpret the Boolean constant ${?}^{TT}($top$)$ as an arbitrary subset of $A.$

interpret the Boolean constant ${?}_{|}{?}_{?}($bottom$)$ as the empty set.